Decomposition of rational maps by stable multicurves

TL;DR

This paper explores how stable multicurves decompose post-critically finite rational maps, revealing conditions for Julia set disjointness, renormalizability, and Fatou domain properties, with implications for understanding complex dynamics.

Contribution

It introduces a new decomposition method based on stable multicurves and establishes criteria for Julia set separation and renormalizability in rational maps.

Findings

Disjointness of small Julia sets characterized by coiling curves

Post-critically finite maps with coiling curves are renormalizable

Provides examples of rational maps with specific Fatou domain properties

Abstract

A completely stable multicurve of a post-critically finite rational map induces a combinatorial decomposition. The projections of the small Julia sets are immersed within the original Julia set. We prove that two small Julia sets are disjoint if and only if they are separated by a coiling curve. Furthermore, we prove that a post-critically finite rational map with a coiling curve is renormalizable. Using a similar argument, we give a sufficient condition for a Fatou domain to qualify as a Jordan domain. By tuning polynomails in such a Fatou domain, we provide examples of post-critically finite rational maps with coiling curves.